International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 521-526

Table 3.4.3.6 

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail:  janovec@fzu.cz

Table 3.4.3.6| top | pdf |
Ferroelastic domain pairs and twins with compatible domain walls

[F_1]: symmetry of domain state [{\bf S}_1]; [g_{1j}]: switching operation, [g_{1j}{\bf S}_1={\bf S}_j]; [K(F_1,g_{1j})]: twinning group, group extension of [F_1] by [g_{1j}]; Axis h: intersection of compatible walls; Equation: component B expressed as a function of strain components or lattice parameters (see end of table); Wall normals: coordinates of normals n1 and n2 of two perpendicular compatible walls, subscript e: wall is charged (see Explanation[link]); [\omega]: obliquity, for numbers (n) see end of table; [{\sf \overline J}_{1j} ]: extended layer-group symmetry of the twin and the wall; [{\underline t}_{1j}^{\star}]: non-trivial twinning operation of the twin; [{\sf T}_{1j}]: layer-group symmetry of the twin and the wall, twin law of the ferroelastic twin; Classification: classification of the twin and the wall (see Table 3.4.4.3[link]).

[F_1][g_{1j}][K(F_1, g_{1j} )]Axis hEquationWall normals n[\omega][{\sf \overline J}_{1j}][\underline t^{\star}_{1j}][{\sf T}_{1j}]Classification
[1] [2^{\star}_z] [2^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2^{\star}_z]   1 [{\rm AR}^{\star}]
[[1B0]_e] [\underline2^{\star}_z] [\underline 2^{\star}_z] [\underline 2^{\star}_z] SI
[1] [m^{\star}_z] [m^{\star}_z] [[B\bar{1}0]] (a)   [[001]_e] (1) [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] SI
[[1B0]] [m^{\star}_z]   1 [{\rm AR}^{\star}]
[\bar{1}] [m^{\star}_z, 2^{\star}_z] [2^{\star}_z/m^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2_z^{\star}/\underline{m}^{\star}_z] [\underline{m}^{\star}_z] [\underline{m}^{\star}_z] SR
[[1B0]] [\underline2^{\star}_z/m^{\star}_z] [\underline2^{\star}_z] [\underline2^{\star}_z] SR
[2_z] [2^{\star}_x,2^{\star}_y] [2^{\star}_x2^{\star}_y2_z] [[001]]     [[100]] (2) [2^{\star}_x\underline2^{\star}_y\underline2_z] [\underline2^{\star}_y] [\underline2^{\star}_y] SR
[[010]] [\underline2^{\star}_x2^{\star}_y\underline2_z] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[2_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_y2_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_y\underline2_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_y\underline2_z] [\underline{m}^{\star}_y] [\underline{m}^{\star}_y] SR
[2_z] [4^{\star}_z,4_z^{3 \star}] [4^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar4^{\star}_z,\bar4_z^{*3}] [\bar{4}^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [3_z,6_z^5] [6_{z}] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[3_z^2,6_z] [6_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar3_z^5,\bar6_z] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[\bar3_z,\bar6_z^5] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_x] [2^{\star}_{xy},4_z] [4_z2_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[2_x] [m^{\star}_{xy},\bar4_z] [\bar4_z2_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{x'},3_z^2] [3_z2_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[2_x] [m^{\star}_{x'},\bar3_z^5] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{y'},6_z] [6_z2_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (7) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[2_x] [m^{\star}_{y'},\bar6_z] [\bar{6}_z2_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (7) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[2_{xy}] [m^{\star}_{x},\bar4_z^{3}] [\bar4_zm_x2_{xy}] [[0B\bar{1}]] (g)   [[100]_e] (8) [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SI
[[01B]] [m^{\star}_{x}]   1 [{\rm AR}^{\star}]
[m_z] [m^{\star}_x,2^{\star}_y] [m^{\star}_x2^{\star}_ym_z] [[001]]     [[100]_e] (2) [\underline{m}^{\star}_x\underline2^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SI
[[010]] [m_x^{\star}2_y^{\star}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{z}] [4_z,\bar4_z^3] [4_z/m_{z}] [[001]] (b)   [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[4_z^3,\bar4_z] [4_z/m_z] [[001]] (b) [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [3_z,\bar6^5_z] [\bar6_{z}] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[3^2_z,\bar6_z] [\bar6_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [\bar3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[\bar3_z^5,6_z] [6_z/m_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_x] [m^{\star}_{xy},4_z] [4_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{xy},\bar4_z] [\bar{4}_zm_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[m_x] [m^{\star}_{x'},3_z^2] [3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{x'},\bar3_z^5] [\bar3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[m_x] [m^{\star}_{y'},6_z] [6_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (6) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{y'},\bar6_z] [\bar{6}_zm_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[m_{xy}] [2^{\star}_{x},\bar4^3_z] [\bar{4}_z2_xm_{xy}] [[0B\bar{1}]] (h)   [[100]] (9) [2^{\star}_{x}]   1 [{\rm AR}^{\star}]
[[01B]_e] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SI
[2_z/m_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_ym_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_ym_z] [\underline{m}^{\star}_y] [\underline2^{\star}_x\underline{m}^{\star}_ym_z] SR
[2_z/m_z] [4^{\star}_z,4^{3 \star}_z] [4^{\star}_z/m_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_z/m_z] [3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[3_z^2,6_z] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_x/m_x] [m^{\star}_{xy},4_z] [4_z/m_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar{1}B]] [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x/m_x] [m^{\star}_{x'},3_z^2] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}/\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}1B]] [\underline2^{\star}_{x'}/m^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x/m_x] [m^{\star}_{y'},6_z] [6_z/m_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}/\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[[\bar1\sqrt{3}B]] [\underline2^{\star}_{y'}/m^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[2_x2_y2_z] [2^{\star}_{x\bar{y}}, 2^{\star}_{xy}] [4^{\star}_z2_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x2_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [\bar{4}^{\star}_z2_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [\underline{m}^{\star}_{xy}m^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [m^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[2_x2_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z2_x2_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x2_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[2_{x\bar{y}}2_{xy}2_z] [m^{\star}_{x},m^{\star}_{y}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[100]] (13) [\underline{m}^{\star}_{x}m^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SR
[[010]] [m^{\star}_{x}\underline{m}^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{y}] [\underline{m}^{\star}_{y}] SR
[2_{x\bar{y}}2_{xy}2_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[2_{x\bar{y}}2_{xy}2_z] [m_{xz}^{\star},\bar4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] SI
[[\bar{1}B1]] [m_{xz}^{\star}]   1 [{\rm AR}^{\star}]
[m_xm_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [4^{\star}_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[m_xm_y2_z] [2^{\star}_{x\bar{y}},2^{\star}_{xy}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[m_xm_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[m_xm_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[m_x2_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [\bar{6}_zm_x2_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [{m}^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[2_xm_ym_z] [m^{\star}_{x\bar{y}},2^{\star}_{xy}] [4_z/m_zm_xm_{xy}] [[001]]     [[110]] (11) [2^{\star}_{xy}m^{\star}_{x\bar{y}}m_z]   [m_z] [{\rm AR}^{\star}]
[[1\bar10]_e] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] SI
[2_xm_ym_z] [m^{\star}_{y'},2^{\star}_{x'}] [\bar{6}_z2_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}m^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[[\sqrt{3}10]_e] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SI
[2_xm_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [m^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{x},2^{\star}_{y}] [\bar{4}^{\star}_z2^{\star}_xm_{xy}] [[001]]   [\Bigl[] [[100]] (13) [2^{\star}_{x}\underline2^{\star}_{y}\underline2_z] [\underline2^{\star}_{y}] [\underline2^{\star}_{y}] SR
[[010]] [\underline2^{\star}_{x}2^{\star}_{y}\underline2_z] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SR
[m_{x\bar{y}}m_{xy}2_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_{x\bar{y}}2_{xy}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}(m^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}2_{xy}m_z] [2^{\star}_{xz},\bar4_y] [m_z\bar{3}_{p}m_{xy}(2^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_xm_ym_z] [m^{\star}_{xy},m^{\star}_{x\bar{y}}] [4^{\star}_z/m_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (10) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{xy}] [\underline2^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}\underline{2}^{\star}_{xy}m_z] SR
[m_xm_ym_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SR
[[\sqrt{3}10]] [m^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}^{\star}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SR
[m_{xy}m_{\bar{x}y}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar{1}B1]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[4_z] [m^{\star}_{xz},\bar{4}_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]_e] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[010]]     [[101]] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] SI
[4_z/m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [4_z3_{p}2_{xy}] [[010]]   [\Bigl[] [[101]] (14) [2^{\star}_{xz}\underline2^{\star}_{x\bar{z}}\underline2_{y}] [\underline2^{\star}_{\bar{x}z}] [\underline2^{\star}_{\bar{x}z}] SR
[[\bar101]] [\underline2^{\star}_{xz}2^{\star}_{x\bar{z}}\underline2_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[4_zm_xm_{xy}] [m^{\star}_{x\bar{z}},2^{\star}_{xz}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SI
[\bar{4}_z2_xm_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [\bar{4}_z3_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[\bar{4}_zm_x2_{xy}] [m_{x\bar{z}}^{\star},2_{xz}^{\star}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[\bar{4}_z2_xm_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}2^{\star}_{x\bar{z}}2_y] [\underline2^{\star}_{x\bar{z}}] [\underline2^{\star}_{x\bar{z}}] SR
[[\bar101]] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] [\underline{2}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] SI
[4_z/m_zm_xm_{xy}] [m^{\star}_{xz},m^{\star}_{\bar{x}z}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{\bar{x}z}m_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}\underline{2}^{\star}_{\bar{x}z}m_y] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[3_{p}] [2^{\star}_{x},3_r] [2_z 3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x]   1 [{\rm AR}^{\star}]
[[011]_e] [\underline2^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SI
[3_{p}] [m^{\star}_{x},\bar3_r] [m_z \bar3_{p}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SI
[[011]] [m^{\star}_x]   1 [{\rm AR}^{\star}]
[3_{p}] [2^{\star}_{xy},4_y] [4_z 3_{p} 2_{xy}] [[1\bar10]]     [[001]_e] (15) [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[[110]] [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[3_{p}] [m^{\star}_{xy},\bar4_y] [\bar4_z3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[110]_e] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[\bar3_{p}] [m^{\star}_{x},3_r] [m_z\bar3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x/\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [\underline2^{\star}_x/m^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[\bar3_{p}] [m^{\star}_{xy},4_y] [m_z\bar3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[[110]] [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[3_{p} 2_{x\bar{y}}] [2^{\star}_{x},2^{\star}_{yz}] [4_z3_{p} 2_{xy}] [[01\bar1]]     [[100]] (15) [2^{\star}_x\underline2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_{yz}] [\underline2^{\star}_{yz}] SR
[[011]] [\underline2^{\star}_x2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[3_{p} 2_{x\bar{y}}] [m^{\star}_{x},m^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm_{yz}^{\star}\underline2_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}\underline2_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}] SR
[3_{p} m_{x\bar{y}}] [2^{\star}_{x},m^{\star}_{yz}] [\bar4_z3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [m^{\star}_{yz}m_{y\bar{z}}2^{\star}_x]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[[011]_e] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] SI
[3_{p} m_{x\bar{y}}] [m^{\star}_x,2^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SI
[[011]] [m^{\star}_x2^{\star}_{yz}m_{y\bar{z}}]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[\bar3_{p}m_{x\bar{y}}] [m^{\star}_x,m^{\star}_{yz}] [m_z \bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline2^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] SR

Expressions for obliquity [\omega] as a function of spontaneous strain components and lattice parameters

 [\omega] as a function of spontaneous strain components [\omega] as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
Expression[\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}] [Scheme scheme1]
(1) [\omega= 2 \sqrt{t^2+u^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\gamma }} \right|]
(2) [\omega= 2|v|] [\omega= |\pi /2 -\gamma |]
(3) [\omega=\sqrt{(q-r)^2+4v^2}] [\omega=\left |\arcsin {{\displaystyle \sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}\right|]
(4) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4v^2}] [\omega= \left|\pi/2-\psi_1- \psi_2\right|]
    [\psi _1={\rm arccotan}{{\displaystyle c^2(a^2+b^2–2d^2)-(a^2-b^2)^2+D(a^2-b^2-d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
    [\psi_2={\rm arccotan}{{\displaystyle b^2(a^2-b^2)+d^2(a^2-b^2)-D(a^2-b^2+d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
    [D=\sqrt{(a^2-d^2)^2-(a^2-b^2)(b^2-d^2)}]
(5) [\omega=\sqrt{(q-r)^2+2t^2}] [\omega=\left|\arcsin {{\displaystyle c^2(a^2+b^2)\sin^2 \alpha -b^2(a+c\cos\alpha)(2Da+c\cos\alpha)}\over {\displaystyle \sqrt{c^2(a^2+b^2) \sin^2\alpha +b^2(a+c\cos\alpha )^2} \sqrt{ c^2(a^2+b^2)\sin^2\alpha+b^2(2Da+c\cos\alpha)^2}}}\right|]
    [D={{\displaystyle ac\cos\alpha }\over {\displaystyle b^2-a^2}}]
(6) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle (4a^2-b^2)\left[\left(1-{ {c\cos\beta }\over{a+b}}\right)b\cos\beta -{{c}\over{2}}\right]+{{3cb^2\sin^2\beta }\over{2}}}\over {\displaystyle \sqrt{4a^2-b^2(1–9\sin^2\beta )} \sqrt {(ac\sin\beta)^2+(4a^2-b^2)\left[1-{{c\cos\beta }\over{a+b}})b-{{c\cos\beta }\over{2}}\right]^2}}}\right| \quad(^*)]
(7) [\omega=2|t|] [\omega=|\pi /2 -\alpha |]
(8) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle 3b^2c-c(4a^2-b^2)\sin^2\alpha +2 b^2D\sqrt{4a^2-b^2}\cos\alpha }\over {\displaystyle \sqrt{b^2+(4a^2-b^2)\sin^2\alpha }\sqrt{ 9b^2c^2+(4a^2-b^2)(c^2\sin^2\alpha +4b^2D^2)+12b^2Dc\sqrt{4a^2-b^2}}}} \right|\quad(^*)]
    [D={{\displaystyle 2ac\cos\alpha }\over {\displaystyle b^2-a^2}} ]
(9) [\omega=2\sqrt{q^2+f^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\alpha }}\right|]
(10) [\omega={{\sqrt3}\over{2}} |q-r| ] [\omega=\left|\arcsin{{\displaystyle b^2-a^2}\over{\displaystyle a\sqrt{2b^2+a^2}}}\right|]
(11) [\omega=|q-r| ] [\omega=\left|\arcsin {{\displaystyle a^2-b^2}\over{\displaystyle b^2+a^2}}\right|]
(12) [\omega=\sqrt{(q-s)^2+2v^2}] [\omega=\left|\arcsin {{\displaystyle c^2(D\cos\gamma -1)-a^2\sin^2\gamma }\over {\displaystyle \sqrt{c^2+a^2 \sin^2\gamma} \sqrt{ 4c^2(D^2-D\cos\gamma )+c^2+a^2\sin^2\gamma }}}\right|]
    [D={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}} ]
(13) [\omega=2|v| ] [\omega= |\pi /2 -\gamma |]
(14) [\omega=|q-s|] [\omega=\left|\arcsin {{\displaystyle a^2-c^2}\over{\displaystyle c^2+a^2}}\right|]
(15) [\omega=2\sqrt2 |v|] [\omega=\left|\arcsin {{\displaystyle \sqrt 2\cos\alpha }\over {\displaystyle 1+\cos\alpha}} \right|]

Expressions for component B of wall normal as a function of spontaneous strain components and lattice parameters

EquationB as a function of spontaneous strain components [\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}]B as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
(a) [B={{\displaystyle t}\over{\displaystyle u}}]  
(b) [B={{\displaystyle 2v+\sqrt{(q-r)^2+4v^2}}\over{\displaystyle q-r}}] [B={{\displaystyle -2ab\cos\gamma +\sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}]
(c) [B={{\displaystyle (q-r)+2\sqrt{3}v+4\sqrt{(q-r)^2+4v^2}}\over{\displaystyle \sqrt3(r-q)+2v}}] [B=2{{\displaystyle a^2-c^2-\sqrt{(a^2-c^2)^2-(a^2-b^2)(b^2-c^2)}}\over{\displaystyle a^2-b^2}}-1]
(d) [B={{\displaystyle 2t}\over{\displaystyle q-r}}]  
(e) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(f) [B={{\displaystyle 4t}\over{\displaystyle q-r}}]  
(g) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(h) [B={{\displaystyle -u}\over {\displaystyle v}}]  
(k) [B={{\displaystyle 2v}\over{\displaystyle s-v}}] [B={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}}]